Damp/wet breeze block wall

I wouldn't recommend it, you will probably not stop the water coming in and just seal water into the wall.. much better to talk to your neighbour and explain maybe they can fit guttering or if not give the outside of the block work a few coats of masonry paint

I would recommend using black jack as water repellent.

I wouldn't recommend any pain of any sort been put on this wall I would ask my neighbour to have a look inside my garage after rain so you can show the neighbour how the water is penetrating through your block work. And then suggest putting gutters up. Then I recommend this is breeze block to use storm dry as this would be absorbed by the block work as blockwork absorbs water however if storm dry is used on this block work when dry they will be no more penetration of water l.

No that a terribly expensive way to solve the issue... Just knock on there door be polite show them the evidence and ask if you can put a gutter up on the shed to redirect the water off the roof. offer to pay for it in full if there decent people there offer to pay for half or all of it. guttering isn’t too expensive to buy it’s labour that expensive if you get a professional in but lead will cost you a fortune and you won’t solve the issue just hide the issue. You should had your breeze block wall rendered then you wouldn’t have an issue at all just a point to note.